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If f is the derivative of F, then we call F an antiderivative of f.
We already know how to find antiderivatives–we just didn't tell you that's what they're called. It's like when you realize what all of the subtle signs in the M. Night Shyamalan movie mean. Seriously, like whoa. Whenever we're given a derivative and we "think backwards" to find a possible original function, we're finding an antiderivative.
Let f(x) = 3x2. Find an antiderivative of f.
We think backwards: what could we take the derivative of to get 3x2? This derivative looks like it came from the power rule, so the original function must involve x3. Since the derivative of x3 is 3x2, the function
F(x) = x3
is an antiderivative of f(x) = 3x2.
Any other antiderivative of 3x2 will have the form x3 + C where C is a constant. We generally take C = 0. For the FTC it won't matter which antiderivative we use, so we might as well use the simplest one.
These exercises should be mostly review, and help you remember how thinking backwards works. You might want to review the rules for taking derivatives first.
To check an answer for this sort of problem, take the derivative of your answer. If you take the derivative of your answer F and get the f given in the problem, then F is an antiderivative of f and you did the problem correctly. Gold stars all around.
Now that we know what antiderivatives are, we can use them along with the FTC to evaluate some integrals we didn't know how to evaluate before. The FTC says that if f is continuous on [a, b] and is the derivative of F, then
This means if we want to know , we
1) find an antiderivative F of f,
2) evaluate F at the limits of integration, and
3) subtract to find F(b) – F(a).
When evaluating definite integrals for practice, you can use your calculator to check the answers. If you don't know how to use your calculator to find integrals you can look in the manual, look online, ask a friend, or ask your teacher. But practice doing integrals by hand until they're so easy you don't even mind anymore.
Here are some reasons to practice doing integrals by hand.
1) At some point you'll probably need to pass a test involving integration, without being allowed to have a calculator. Midterm, anyone?
2) Even when you are allowed a calculator, your teacher will probably want to see the steps you took to get your answer.
3) If you're asked to integrate something that uses letters instead of numbers, the calculator won't help much (some of the fancier calculators will, but see the first two points).
4) Later in Calculus you'll start running into problems that expect you to find an integral first and then do other things with it. It will sometimes be easier to find the integral by hand than it will be to distract yourself by putting the integral into your calculator.
Before asking you to find too many definite integrals, we should share a nice notational shortcut. To abbreviate
F(b) – F(a),
This expression is read "F of x evaluated from a to b."
Using this shortcut, our work to find would look like this:
This is a nice shortcut because it saves us from having to mess with letters like f and F. We just found the antiderivative
put it in brackets
drew a vertical line on the right-hand side
and wrote the limits of integration
Then expand the shortcut
[(2)3] – [(0)3]
and simplify to get the answer.
Remember that when we expand the shortcut, we use the upper limit of integration first:
No, we're not talking about snail mail.
If we have a constant instead of a number for a limit of integration, not much changes. We apply the FTC, and write a constant instead of a number where it's appropriate to do so.
If c is a constant greater than 0, find .
We apply the FTC like always, but use c for the upper limit of integration instead of a number.
Now we can answer questions like this:
If , what is c?
We just worked out that
So if the integral is equal to 2, it means
Solving, we get
c3 = 6
When there's a constant in the integrand, you have to take it into account while finding the antiderivative. If there's a constant in the integrand, that constant will also show up in the antiderivative.
A word of warning: when there are constants in the integrand, it can be easy to get mixed up when it comes time to put in the limits of integration. Do they plug into the a or the x? The answer is that the limits of integration plug into the variable of integration. In this example, the limits of integration (1 and 5) went into the variable of integration x, not into the constant a.
Be Careful: Plug the limits of integration into the variable of integration. If there are constants in the integrand, leave those alone. DO NOT plug the limits of integration into any constants.
Integrals like to flip-flop on their stance from time to time. Seriously, they're as bad as politicians sometimes. Sometimes you think they're left, sometimes you think they're right, sometime the upper limit is smaller than the lower limit...
When we originally stated the FTC we said that if f is continuous on [a, b], then
where F ' = f.
We can still evaluate integrals this way if the upper limit of integration is smaller than the lower limit.
Suppose this is the case, so b < a. By properties of integrals,
Since b < a we can use the FTC to say
Practically speaking, this means you can evaluate integrals without worrying which limit of integration is bigger. The integrand should still be continuous on the interval between the limits of integration, though.
Some say po-TAY-to, some say po-TAH-to. Some say to-MAY-to, some say to-MAH-to. Some choose to not include a constant when finding an antiderivative. Some do include the constant. Here's why it doesn't matter what we do with the constant.
The simplest antiderivative of 4x3 is x4. Using the FTC with that antiderivative, we get
Now let's try the FTC with a different antiderivative. How about x4 + 3?
Notice how the extra "+ 3"s canceled each other out and we got 15 again. If we used some other antiderivative of 4x3, the same sort of thing would happen.
The moral of the story is that when evaluating a definite integral with the FTC, no matter which antiderivative you use, you should get the same answer every time. Since it doesn't matter which antiderivative you use, you may as well use the simplest one.
The average value of the function f on the interval [a,b] is the integral of the function on that interval divided by the length of the interval. Since we know how to find the exact values of a lot of definite integrals now, we can also find a lot of exact average values. What's the average value of an "A" in Calculus class? You tell us.
Find the average value of f(x) = sin x on the interval .
The average value of f(x) = sin x on this interval is
Since we know how to evaluate the integral, we know how to find the average value. First let's simplify that stuff out in front of the integral:
Now we can rewrite the average value to be a little more tidy.
It's tempting to go off and compute the integral in a corner of your paper, then come back and multiply by at the end. Unfortunately, that's dangerous. After working out a long integral, it's very easy to forget to come back and do that last step. Don't do it.